Reference documentation for deal.II version 9.3.0
SolverFIRE< VectorType > Class Template Reference

#include <deal.II/lac/solver_fire.h>

Inheritance diagram for SolverFIRE< VectorType >:
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## Public Types

using vector_type = VectorType

## Public Member Functions

SolverFIRE (SolverControl &solver_control, VectorMemory< VectorType > &vector_memory, const AdditionalData &data=AdditionalData())

template<typename PreconditionerType = DiagonalMatrix<VectorType>>
void solve (const std::function< double(VectorType &, const VectorType &)> &compute, VectorType &x, const PreconditionerType &inverse_mass_matrix)

template<typename MatrixType , typename PreconditionerType >
void solve (const MatrixType &A, VectorType &x, const VectorType &b, const PreconditionerType &preconditioner)

boost::signals2::connection connect (const std::function< SolverControl::State(const unsigned int iteration, const double check_value, const VectorType &current_iterate)> &slot)

template<class Archive >
void serialize (Archive &ar, const unsigned int version)

Subscriptor functionality

Classes derived from Subscriptor provide a facility to subscribe to this object. This is mostly used by the SmartPointer class.

void subscribe (std::atomic< bool > *const validity, const std::string &identifier="") const

void unsubscribe (std::atomic< bool > *const validity, const std::string &identifier="") const

unsigned int n_subscriptions () const

template<typename StreamType >
void list_subscribers (StreamType &stream) const

void list_subscribers () const

## Static Public Member Functions

static ::ExceptionBaseExcInUse (int arg1, std::string arg2, std::string arg3)

static ::ExceptionBaseExcNoSubscriber (std::string arg1, std::string arg2)

## Protected Member Functions

virtual void print_vectors (const unsigned int, const VectorType &x, const VectorType &v, const VectorType &g) const

## Protected Attributes

GrowingVectorMemory< VectorTypestatic_vector_memory

VectorMemory< VectorType > & memory

boost::signals2::signal< SolverControl::State(const unsigned int iteration, const double check_value, const VectorType &current_iterate), StateCombineriteration_status

## Detailed Description

### template<typename VectorType = Vector<double>> class SolverFIRE< VectorType >

FIRE (Fast Inertial Relaxation Engine) for minimization of (potentially non-linear) objective function $$E(\mathbf x)$$, $$\mathbf x$$ is a vector of $$n$$ variables ( $$n$$ is the number of variables of the objective function). Like all other solver classes, it can work on any kind of vector and matrix as long as they satisfy certain requirements (for the requirements on matrices and vectors in order to work with this class, see the documentation of the Solver base class). The type of the solution vector must be passed as template argument, and defaults to Vector<double>.

FIRE is a damped dynamics method described in Structural Relaxation Made Simple by Bitzek et al. 2006, typically used to find stable equilibrium configurations of atomistic systems in computational material science. Starting from a given initial configuration of the atomistic system, the algorithm relies on inertia to obtain (nearest) configuration with least potential energy.

Notation:

• The global vector of unknown variables: $$\mathbf x$$.
• Objective function: $$E(\mathbf x)$$.
• Rate of change of unknowns: $$\mathbf v$$.
• Gradient of the objective function w.r.t unknowns: $$\mathbf g = \nabla E(\mathbf x)$$.
• Mass matrix: $$\mathbf M$$.
• Initial guess of unknowns: $$\mathbf x_0$$.
• Time step: $$\Delta t$$.

Given initial values for $$\Delta t$$, $$\alpha = \alpha_0$$, $$\epsilon$$, $$\mathbf x = \mathbf x_0$$ and $$\mathbf v= \mathbf 0$$ along with a given mass matrix $$\mathbf M$$, FIRE algorithm is as follows,

1. Calculate $$\mathbf g = \nabla E(\mathbf x)$$ and check for convergence ( $$\mathbf g \cdot \mathbf g < \epsilon^2$$).
2. Update $$\mathbf x$$ and $$V$$ using simple (forward) Euler integration step,
$$\mathbf x = \mathbf x + \Delta t \mathbf v$$,
$$\mathbf v = \mathbf v + \Delta t \mathbf M^{-1} \cdot \mathbf g$$.
3. Calculate $$p = \mathbf g \cdot \mathbf v$$.
4. Set $$\mathbf v = (1-\alpha) \mathbf v + \alpha \frac{|\mathbf v|}{|\mathbf g|} \mathbf g$$.
5. If $$p<0$$ and number of steps since $$p$$ was last negative is larger than certain value, then increase time step $$\Delta t$$ and decrease $$\alpha$$.
6. If $$p>0$$, then decrease the time step, freeze the system i.e., $$\mathbf v = \mathbf 0$$ and reset $$\alpha = \alpha_0$$.

Also see Energy-Minimization in Atomic-to-Continuum Scale-Bridging Methods by Eidel et al. 2011.

Definition at line 89 of file solver_fire.h.

## ◆ vector_type

template<class VectorType = Vector<double>>
 using SolverBase< VectorType >::vector_type = VectorType
inherited

An alias for the underlying vector type

Definition at line 347 of file solver.h.

## ◆ SolverFIRE() [1/2]

template<typename VectorType = Vector<double>>
 SolverFIRE< VectorType >::SolverFIRE ( SolverControl & solver_control, VectorMemory< VectorType > & vector_memory, const AdditionalData & data = AdditionalData() )

Constructor.

## ◆ SolverFIRE() [2/2]

template<typename VectorType = Vector<double>>
 SolverFIRE< VectorType >::SolverFIRE ( SolverControl & solver_control, const AdditionalData & data = AdditionalData() )

Constructor. Use an object of type GrowingVectorMemory as a default to allocate memory.

## ◆ solve() [1/2]

template<typename VectorType = Vector<double>>
template<typename PreconditionerType = DiagonalMatrix<VectorType>>
 void SolverFIRE< VectorType >::solve ( const std::function< double(VectorType &, const VectorType &)> & compute, VectorType & x, const PreconditionerType & inverse_mass_matrix )

Obtain a set of variables x that minimize an objective function described by the polymorphic function wrapper compute, with a given preconditioner inverse_mass_matrix and initial x values. The function compute returns the objective function's value and updates the objective function's gradient (with respect to the variables) when passed in as first argument based on the second argument– the state of variables.

## ◆ solve() [2/2]

template<typename VectorType = Vector<double>>
template<typename MatrixType , typename PreconditionerType >
 void SolverFIRE< VectorType >::solve ( const MatrixType & A, VectorType & x, const VectorType & b, const PreconditionerType & preconditioner )

Solve for x that minimizes $$E(\mathbf x)$$ for the special case when $$E(\mathbf x) = \frac{1}{2} \mathbf x^{T} \mathbf A \mathbf x - \mathbf x^{T} \mathbf b$$.

## ◆ print_vectors()

template<typename VectorType = Vector<double>>
 virtual void SolverFIRE< VectorType >::print_vectors ( const unsigned int, const VectorType & x, const VectorType & v, const VectorType & g ) const
protectedvirtual

Interface for derived class. This function gets the current iteration x (variables), v (x's time derivative) and g (the gradient) in each step. It can be used for graphical output of the convergence history.

## ◆ connect()

template<class VectorType >
 boost::signals2::connection SolverBase< VectorType >::connect ( const std::function< SolverControl::State(const unsigned int iteration, const double check_value, const VectorType ¤t_iterate)> & slot )
inlineinherited

Connect a function object that will be called periodically within iterative solvers. This function is used to attach monitors to iterative solvers, either to determine when convergence has happened, or simply to observe the progress of an iteration. See the documentation of this class for more information.

Parameters
 slot A function object specified here will, with each call, receive the number of the current iteration, the value that is used to check for convergence (typically the residual of the current iterate with respect to the linear system to be solved) and the currently best available guess for the current iterate. Note that some solvers do not update the approximate solution in every iteration but only after convergence or failure has been determined (GMRES is an example); in such cases, the vector passed as the last argument to the signal is simply the best approximate at the time the signal is called, but not the vector that will be returned if the signal's return value indicates that the iteration should be terminated. The function object must return a SolverControl::State value that indicates whether the iteration should continue, has failed, or has succeeded. The results of all connected functions will then be combined to determine what should happen with the iteration.
Returns
A connection object that represents the connection from the signal to the function object. It can be used to disconnect the function object again from the signal. See the documentation of the BOOST Signals2 library for more information on connection management.

Definition at line 554 of file solver.h.

## ◆ subscribe()

 void Subscriptor::subscribe ( std::atomic< bool > *const validity, const std::string & identifier = "" ) const
inherited

Subscribes a user of the object by storing the pointer validity. The subscriber may be identified by text supplied as identifier.

Definition at line 136 of file subscriptor.cc.

## ◆ unsubscribe()

 void Subscriptor::unsubscribe ( std::atomic< bool > *const validity, const std::string & identifier = "" ) const
inherited

Unsubscribes a user from the object.

Note
The identifier and the validity pointer must be the same as the one supplied to subscribe().

Definition at line 156 of file subscriptor.cc.

## ◆ n_subscriptions()

 unsigned int Subscriptor::n_subscriptions ( ) const
inlineinherited

Return the present number of subscriptions to this object. This allows to use this class for reference counted lifetime determination where the last one to unsubscribe also deletes the object.

Definition at line 301 of file subscriptor.h.

## ◆ list_subscribers() [1/2]

template<typename StreamType >
 void Subscriptor::list_subscribers ( StreamType & stream ) const
inlineinherited

List the subscribers to the input stream.

Definition at line 318 of file subscriptor.h.

## ◆ list_subscribers() [2/2]

 void Subscriptor::list_subscribers ( ) const
inherited

List the subscribers to deallog.

Definition at line 204 of file subscriptor.cc.

## ◆ serialize()

template<class Archive >
 void Subscriptor::serialize ( Archive & ar, const unsigned int version )
inlineinherited

Read or write the data of this object to or from a stream for the purpose of serialization using the BOOST serialization library.

This function does not actually serialize any of the member variables of this class. The reason is that what this class stores is only who subscribes to this object, but who does so at the time of storing the contents of this object does not necessarily have anything to do with who subscribes to the object when it is restored. Consequently, we do not want to overwrite the subscribers at the time of restoring, and then there is no reason to write the subscribers out in the first place.

Definition at line 310 of file subscriptor.h.

## Member Data Documentation

template<typename VectorType = Vector<double>>
protected

Additional data to the solver.

Definition at line 179 of file solver_fire.h.

## ◆ static_vector_memory

template<class VectorType = Vector<double>>
 GrowingVectorMemory SolverBase< VectorType >::static_vector_memory
mutableprotectedinherited

A static vector memory object to be used whenever no such object has been given to the constructor.

Definition at line 415 of file solver.h.

## ◆ memory

template<class VectorType = Vector<double>>
 VectorMemory& SolverBase< VectorType >::memory
protectedinherited

A reference to an object that provides memory for auxiliary vectors.

Definition at line 420 of file solver.h.

## ◆ iteration_status

template<class VectorType = Vector<double>>
 boost::signals2::signal< SolverControl::State(const unsigned int iteration, const double check_value, const VectorType & current_iterate), StateCombiner> SolverBase< VectorType >::iteration_status
protectedinherited

A signal that iterative solvers can execute at the end of every iteration (or in an otherwise periodic fashion) to find out whether we should continue iterating or not. The signal may call one or more slots that each will make this determination by themselves, and the result over all slots (function calls) will be determined by the StateCombiner object.

The arguments passed to the signal are (i) the number of the current iteration; (ii) the value that is used to determine convergence (oftentimes the residual, but in other cases other quantities may be used as long as they converge to zero as the iterate approaches the solution of the linear system); and (iii) a vector that corresponds to the current best guess for the solution at the point where the signal is called. Note that some solvers do not update the approximate solution in every iteration but only after convergence or failure has been determined (GMRES is an example); in such cases, the vector passed as the last argument to the signal is simply the best approximate at the time the signal is called, but not the vector that will be returned if the signal's return value indicates that the iteration should be terminated.

Definition at line 471 of file solver.h.

The documentation for this class was generated from the following file: